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arxiv: 2605.31441 · v1 · pith:WWXM43NInew · submitted 2026-05-29 · 🪐 quant-ph

Intrinsic locality dimension of quantum codes

Yimin Lu , Esther Xiaozhen Fu , Zi-Wen Liu This is my paper

Pith reviewed 2026-06-28 22:05 UTC · model grok-4.3

classification 🪐 quant-ph
keywords quantum error-correcting codesstabilizer codesintrinsic locality dimensionfault-tolerant gatestopological codesalgebraic codesself-correcting memories
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The pith

Stabilizer quantum codes possess an intrinsic locality dimension that limits their parameters and compatible fault-tolerant gates without reference to any fixed background geometry.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper defines an intrinsic locality dimension for stabilizer codes by adapting tools from fractal geometry and geometric measure theory. This dimension functions as an organizing parameter that applies uniformly to topological codes, algebraic constructions such as bivariate-bicycle codes, and other flexible architectures, including non-integer values. It yields general bounds on code parameters and on the logical gates that admit fault-tolerant implementations, extending earlier results that assumed regular lattices. The same parameter produces a conditional prohibition on self-correcting quantum memories in any dimension strictly less than three.

Core claim

The intrinsic locality dimension of stabilizer codes, defined independently of background geometry via fractal geometry and geometric measure theory, serves as a fundamental organizing parameter that induces general limitations on code parameters and on the fault-tolerant logical gates compatible with those codes.

What carries the argument

The intrinsic locality dimension, a real-valued locality measure for codes that is independent of embedding geometry and accommodates flexible and non-integer cases.

Load-bearing premise

The intrinsic locality dimension is well-defined for the stabilizer codes under consideration and captures the locality properties that govern their parameters and gates.

What would settle it

A concrete stabilizer code family whose measured intrinsic dimension permits code parameters or logical gates that exceed the derived general bounds.

Figures

Figures reproduced from arXiv: 2605.31441 by Esther Xiaozhen Fu, Yimin Lu, Zi-Wen Liu.

Figure 1
Figure 1. Figure 1: FIG. 1. Assouad dimension. (a) After “zooming out” so that [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Representetive self-similar quasi-convex fractal mod [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Illustration of the mapping from 3D toric code to 2D planar code with long range connectivity: the [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Illustration of the generalized BPT partition. (a) A tri-partition for a triangle. (b) [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Illustration of multi-code-block gates. The red dot [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. The quasi-tube [PITH_FULL_IMAGE:figures/full_fig_p031_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. A logical operator in the fractalized one-dimensional Ising model. Although the operator has fractal support, the [PITH_FULL_IMAGE:figures/full_fig_p035_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. A two-dimensional layout of the fractalized one-dimensional cluster model. Each orange circle denotes one site [PITH_FULL_IMAGE:figures/full_fig_p035_8.png] view at source ↗
read the original abstract

Quantum error-correcting codes are a cornerstone of quantum computing, with broad and profound connections to physics and mathematics. In this work, we introduce the notion of intrinsic locality dimension of stabilizer codes that is independent of any background geometry and naturally incorporates flexible architectures and accommodates noninteger values, drawing on mathematical machinery from fractal geometry and geometric measure theory. Important scenarios include topological codes and algebraic codes such as bivariate-bicycle-type codes. We show how the intrinsic dimension serves as a fundamental organizing parameter that unifies code properties. In particular, we prove general limitations on code parameters and compatible fault-tolerant logical gates induced by the intrinsic dimension, generalizing the Bravyi--Poulin--Terhal and Bravyi--K\"{o}nig bounds for regular topological codes, respectively. Furthermore, we discuss implications on thermal properties, presenting a conditional no-go result for self-correcting quantum memories in dimension $3-\epsilon$ for any $\epsilon>0$. Our theory lays a versatile and unifying mathematical foundation for studying the fundamental capabilities and geometric implementations of quantum error correction and fault tolerance.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript introduces an intrinsic locality dimension for stabilizer codes, defined independently of background geometry via tools from fractal geometry and geometric measure theory. This dimension is positioned as an organizing parameter that unifies code properties across topological and algebraic families (including bivariate-bicycle codes). The central results are general bounds on code parameters and on compatible fault-tolerant logical gates that generalize the Bravyi-Poulin-Terhal and Bravyi-König theorems, respectively, together with a conditional no-go theorem for self-correcting quantum memories in dimension 3-ε.

Significance. If the claimed generalizations hold, the work supplies a dimension-based unification that extends known no-go results beyond regular lattices to non-integer and flexible architectures. The explicit use of geometric measure theory to accommodate non-integer dimensions and the conditional thermal-stability result constitute concrete strengths that could influence both code design and the study of self-correction.

minor comments (2)
  1. The abstract states that the dimension 'naturally incorporates flexible architectures,' but the manuscript should include at least one explicit calculation for a bivariate-bicycle code showing how the new dimension is computed and recovers the expected locality properties.
  2. Notation for the intrinsic dimension (presumably introduced in an early section) should be compared side-by-side with the classical Hausdorff dimension to clarify the precise technical departure.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and for the positive assessment of the potential significance of the intrinsic locality dimension as a unifying parameter. We note that the referee has not raised any specific major comments or questions about the manuscript.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper introduces a new intrinsic locality dimension drawn from established tools in fractal geometry and geometric measure theory, then uses it as an organizing parameter to generalize existing bounds (Bravyi-Poulin-Terhal, Bravyi-König) to broader stabilizer code families. No load-bearing step reduces by construction to a fitted parameter, self-citation chain, or self-referential definition; the derivation chain remains self-contained against external mathematical machinery and does not rename or smuggle prior results via internal citations.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the applicability of fractal geometry and geometric measure theory to define a dimension for stabilizer codes and on the assumption that this dimension controls code parameters and gate sets in the stated way.

axioms (1)
  • domain assumption Mathematical machinery from fractal geometry and geometric measure theory applies directly to stabilizer codes and yields a well-defined intrinsic locality dimension.
    Invoked to introduce the core new object in the abstract.

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